integrating factor

For

$$\frac{dy}{dt}+p(t)y(t)=g(t)$$

consider a $\mu (t)$ that

$$\mu \frac{dy}{dt}+\mu p(t)y(t)=\mu g(t)$$

take $\frac{d\mu }{dt}=\mu p(t)⟹\mu =C{e}^{p(t)t}$
therefore, we can turn the equation above to

$$\frac{d}{dt}(\mu y)=\mu g(t)⟹y=\frac{\int \mu g(t)dt+C}{\mu }$$

then plug $\mu$ in as the solution.